# De Morgan’s Theorem

In this article, we are going to learn about De Morgan’s Theorem in digital electronics according to the theory, and also we will prove the theorems. So let’s start with the definition of De Morgan’s Theorem.

Also Read: Number System In Digital Electronics

## What are De Morgan’s Theorems?

We use De Morgan’s theorems to solve the expressions of Boolean Algebra.

It is a very powerful tool used in digital design. This theorem explains that the complements of the products of all the terms are equal to the sums of the complements of each and every term.

Likewise, the complements of the sums of all the terms are equal to the products of the complements of each and every term.

The two theorems suggested by De Morgan and which are extremely useful in Boolean algebra are as follows:

### Theorem 1 of De Morgan’s Theorem

\boxed{\mathbf{\overline{AB} = \bar A + \bar B : NAND = Bublled \,OR}}

This theorem states that the complement of a product is equal to the addition of the complements.

This rule is illustrated in the below figure. The left-hand side(LHS) of this theorem represents a NAND Gate with inputs A and B whereas the right-hand side (RHS) of the theorem represents an OR Gate with inverted inputs.

This OR Gate is called “Bubbled OR“. Thus we can state De Morgan’s first theorem as,

\boxed{\mathbf{NAND \equiv Bubbled\,OR}}

### Theorem 2 of De Morgan’s Theorem

\boxed{\mathbf{\overline{A + B} = \bar A \cdot \bar B : NOR = Bubbled \, AND}}

This theorem states that the complement of addition is equal to the product of the complements.

This theorem is illustrated in the figure below. The LHS of this theorem represents a NOR Gate with inputs A and B whereas the RHS represents an AND Gate with inverted inputs.

This AND Gate is called “Bublled AND”. Thus we can state De Morgan’s second theorem as:

\boxed{\mathbf{NOR \equiv Bubbled\,AND}}

This theorem can be verified by writing a truth table for both sides of the theorem statement. This truth table is shown in the figure which shows that LHS = RHS.

## FAQs

### What is De Morgan known for?

De Morgan, a prominent mathematician from Cambridge, is recognized for his insights into algebra as a purely symbolic field. He acknowledged the potential existence of algebras that differ from the standard kind.

### What do De Morgan’s theorems imply?

De Morgan’s theorems have applications in demonstrating that a NAND gate is equivalent to an OR gate with inverted inputs, and a NOR gate is equivalent to an AND gate with inverted inputs. For simplifying expressions with extensive bars, it’s necessary to break down these bars first.

### What are the fundamental properties of Boolean algebra?

The core properties of Boolean algebra include commutativity, associativity, and distributivity.

### What do De Morgan’s theorems state?

De Morgan’s theorems are employed to solve problems in Boolean Algebra, playing a vital role in digital design. These theorems explain that if we want to find the opposite of the combined products of all terms, it’s the same as finding the combined sums of the opposites for each term. Similarly, if we want to find the opposite of the combined sums of all terms, it’s the same as finding the combined products of the opposites for each term.  Join Our Telegram Group!